order statistics subscripts

Hi,
Yes, k:n is the kth order statistic out of n, and k-2:n the (k-2)th order statistic out of n. The final answer should depend on 2, not on k. I guess it will integrate out once I get the right expression.
Thanks!

transformation

Matheagle,

Thanks. My 2-2 transformation ends up looking very similar. I set one variable (I'll rename them from my original post so we don't run into confusion with the variables that you use) to be
and I then get that

and The Jacobian turns out to be equal to 1 in that transformation and I plug in the new variables and get
I'm not sure how to manipulate the variables to get it into the form of a beta eventually (before or after integration).

basic calculus

Thanks! I do owe you. I didn't see the forest for the trees (or maybe I banged my head against the wall one too many times).

I found another reference as well (Order Statistics, 3rd ed., David & Nagajara) - they use

t = v * (1-s) and integrate over v.

I'll still have to mull it over for a while on how I can get it to be more intuitive. (Still don't quite get the beta with the easier substitution since the s outside the integral comes back into the integral for me when I do this.)

you can read this as wellSpringerLink - Journal Article
I wanted you to try that other substitution
and see if you obtained the same result I did.
After dinner I may post my work.............

BACK to my variables... we have

Let s=y-z but now t=z, the smaller of the two order stats. The jacobian should be one again
That doesn't need checking in these cases.
The density of these two is...

That region becomes t>0, s>0 and s+t<1

So the density of s becomes

NOW use calc one, to make the bounds go from 0 to 1, let

which gives you and

and all I see are beta constants now...........

It looks like your reference did what I did.
I have David's book somewhere in my office, but I usually derive these things from scratch.
I can do most things faster by myself than looking it up.
Usually I can't find it or it may never have been done before.
So I just do it.

You're welcome, this is what I do for a living
(if you saw that paper).
I was trying to help another person this morning and he became so rude.
I don't have to do this for free.
I gave him plenty of help and I won't ever again.
All I expect is a thank you.